I'm trying to make a vertex group on one side of a model, but the selection keeps mirroring on the other side too. How do I stop this?
The Mirror Modifier is a non-destructive action within Blender. Meaning the changes are virtual and the results do not make any permanent changes.


(This is just a UV sphere with half the vertices deleted and then a Mirror Modifier added.)



When you edit one side of the Mesh a representation of the same change is shown in the mirrored vertices.

If you want the vertices to exist independantly of the Modifier then you must click Apply.

 

Now the Modifier is applied any changes apply to only the selected vertices. 



The same is true of Vertex groups. As the geometry only exists in one place (i.e. the left) when the Modifier is working. As soon as you apply it you can affect any of the now real vertices.